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Regular Compactifications of Convergence Spaces

1972; American Mathematical Society; Volume: 31; Issue: 2 Linguagem: Inglês

10.2307/2037573

1088-6826

Geoffrey Richardson, D. C. Kent,

Approximation Theory and Sequence Spaces

This note gives a simple characterization for the class of convergence spaces for which regular compactifications exist and shows that each such convergence space has a largest regular compactification. Introduction.It has been shown by Wyler [5] that for every Hausdorff convergence space S there is a regular (including Hausdorff) compact convergence space S* and a continuous map j:S-+S* with the following property: for every continuous map f-.S-^-T, where Pis regular and compact, there is a unique continuous map g:S*^>-T such that f=g°j-Richardson [4] obtained a similar result, but with the following important distinctions: (1) the compactification space S* is Hausdorff but not necessarily regular (for convergence spaces, Hausdorff plus compact does not imply regular); (2) the map j is a dense embedding.But there is in general no largest Hausdorff compactification, and indeed the number of distinct maximal Hausdorff compactifications can be quite large.The conclusions of both [4] and [5] suggest that a more satisfactory compactification theory for convergence spaces might result from an investigation of regular compactifications, although it is known (see [2]) that there are regular convergence spaces which cannot be embedded in any compact regular space.What we obtain in this note is a characterization of the class of convergence spaces for which regular compactifications exist, and we show that each such convergence space has a largest regular compactification.